How to Simulate Rasch Data

1. Decide about the items. They are usually uniformly distributed. How many items? How wide the interval? The item mean is usually set at 0 logits. Simulate the item difficulties.

2. Decide about the person sample. This is usually normally distributed. How big a sample? What is the mean? What is the standard deviation? Simulate the person abilities.

4. For each response by a person to an item:

4A. Generate a random number U = uniform [0,1]

4B. Probability of failure = 1/(1 + exp(ability - difficulty))

4C. If U > Probability of failure, then X=1 else X=0.

4D. X is the simulated observation.

5. Check this by simulating data for a very high ability person (logit = 10): the data should all be "1".
Simulate data for a very low ability person (logit = -10): the data should all be "0"

Polytomous (rating scale or partial credit) data:

1. Decide about the items. They are usually uniformly distributed. How many items? How wide the interval? The item mean is usually set at 0 logits. Simulate the item difficulties.

2. Decide about the person sample. This is usually normally distributed. How big a sample? What is the mean? What is the standard deviation? Simulate the person abilities.

3. Decide about the number of categories, m. The higher categories, 2 to m, have Rasch-Andrich threshold values that are usually ascending and sum to zero across all the categories. Simulate the threshold values.

5. Check this by simulating data for a very high ability person (logit = 10): the data should all be "m" (the top category).
Simulate data for a very low ability person (logit = -10): the data should all be "1" (the bottom category).

Unobserved Categories: sampling zeroes

Unobserved categories have a very low probability of being observed, so set the threshold values:
very low for an unobserved bottom category: example: 5 categories, bottom category unobserved: -40, -1, 0, 1
very high for an unobserved top category: example: 5 categories top category unobserved: -1, 0, 1, 40
very high then very low for the an unobserved intermediate category: example: 5 categories middle category unobserved: -1, 40, -40, 1

Many-Facets data:

As Polytomous data with the addition of:
1A. Decide about the other facets (tasks, demographics, etc.). Choose logit values for their elements.

Go to Institute for Objective Measurement Home Page.
The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website, www.rasch.org.